Understanding z-scores

Z-scores tell us how many standard deviations a data point is from the mean. A number further away from 0 in either direction indicates a data point further from the mean. Z-scores use the normal distribution discussed in the previous section.

Remember the normal distribution?

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The image above shows z-scores. Z-scores help us understand how a specific value compares to the average value in a set of data. By converting any data distribution into z-scores, we standardise the values, making the average (mean) equal to 0.

• Negative z-scores: These indicate values below the mean.
• Positive z-scores: These indicate values above the mean.

Example:

Imagine we have test scores from a class, and the average score is 70. If a student scored 80, their z-score might be +1. This means their score is 1 standard unit above the average. If another student scored 60, their z-score might be -1, meaning their score is 1 standard unit below the average.

In summary, z-scores show us how far and in what direction a value is from the average.

Recap: Mean and Standard Deviation

The mean is the sum of all values divided by the number of values.

The standard deviation tells us the extent a population or sample differs from the mean. Standard deviations use the original measure – e.g., age, height etc. This makes it difficult to compare different variables.

Importance of Z-Scores

Each specific data point will be a certain distance away from the average. Some values may be outliers, far from the average, or close by. Sometimes we want to calculate a definitive number for this, and that’s where we bring in the Z-Score.

As stated, Z-Scores tell us a data point’s distance from the mean. This is useful for talking about outliers, where the Z-Score will be wildly off what you typically observe. You can also identify maximum values, which will have large, positive Z-Scores.

Z-Tables

Z-scores are also used to find probabilities without having to perform any calculations. A computer usually does this for you. However, understanding Z-Tables (link – further reading) can be useful. These tables also allow you to work out where individuals are on the distribution.

A Z-Table is laid out in rows and columns. The rows correspond to the first decimal place, and the columns to the second decimal place. A Z-Table will be included in the files for this section and will be there for you to look at for the tasks.

We can work backwards from a Z-Table too. The number 1.96 is often used in statistics. The Z-Score of 1.96 corresponds to 0.05, or the 5% level.

It is necessary to be able to read a Z-Table, albeit maths adjacent. In the video, a worked example is gone through. At this stage, you do not need to be able to interpret the result in practice. This is because interpretation is largely down to context.